Math 7

Converting square and cubic units

A meter is a hundred centimeters. Why is a square meter not a hundred square centimeters?

You need

Conversion factors and dimensional analysisMath 6 Volume with unit cubesGrade 5

Where you are 6 in Systems of units

Measurement12

Systems of units6

The customary system The metric system and SI units Metric prefixes Conversion factors and dimensional analysis

Converting square and cubic units

Multi-step problems with unit conversions

When you reach for it

You reach for this move whenever an area or a volume is measured in one unit and needed in another. A room is measured in feet but carpet is sold by the square yard. A field is measured in meters but farmland is sold by the hectare. A fish tank is measured in centimeters but its pump is rated in liters.

For a length, the move is the one from conversion factors: since 1\text{ ft} = 12\text{ in}, a length in feet becomes inches by multiplying by 12. Areas and volumes need more care, because an area unit is not a length. A square foot is the area of a square 1 ft on each side, and a cubic yard is the volume of a cube 1 yd on each edge. The question is how many of the small units fit inside one big one, and the figures answer it by counting.

Figure 1A square foot is 12 rows of 12 square inches, so 1\text{ ft}^2 = 12 \times 12 = 144\text{ in}^2.

Each side of a square foot is 12 inches, so the square holds 12 rows of 12 square inches. That is 12 \times 12 = 144 square inches, not 12, for the same reason a rectangle’s area is length times width (see area of rectangles): the factor 12 is used once along each side.

Figure 2A cubic yard is 3 layers of 3 \times 3 = 9 cubic feet, so 1\text{ yd}^3 = 27\text{ ft}^3.

A cube 1 yd on each edge is 3 ft on each edge. Its bottom layer holds 3 \times 3 = 9 cubes of 1 cubic foot, and there are 3 layers, so it holds 27 cubic feet (see volume with unit cubes). The length factor 3 is used three times.

This is the same rule as for scale copies in scale factor, area and volume: multiply every length by k and the area is multiplied by k^2, the volume by k^3. Changing the unit is a scaling of that kind.

The procedure

The unit is a product of lengths

The notation says what to do. A square foot is written \text{ft}^2 because it is a foot times a foot, and units multiply the way numbers do. Replace each foot with 12 inches:

\begin{aligned} 1\text{ ft}^2 &= 1\text{ ft} \times 1\text{ ft} \\ &= 12\text{ in} \times 12\text{ in} \\ &= 144\text{ in}^2 \end{aligned}

In the same way a cubic meter is a meter three times over:

\begin{aligned} 1\text{ m}^3 &= 100\text{ cm} \times 100\text{ cm} \times 100\text{ cm} \\ &= 1{,}000{,}000\text{ cm}^3 \end{aligned}

So the length factor is used twice for an area unit and three times for a volume unit: squared and cubed.

Multiplying by one, as many times as the unit needs

Since 12\text{ in} = 1\text{ ft}, the fraction \displaystyle \frac{12\text{ in}}{1\text{ ft}} equals 1, and multiplying by 1 changes only the units an amount is written in. An area in square feet carries the foot twice, so the fraction is used twice, once to cancel each foot. For 3 square feet:

\begin{aligned} 3\text{ ft}^2 &= 3\text{ ft}^2 \times \frac{12\text{ in}}{1\text{ ft}} \times \frac{12\text{ in}}{1\text{ ft}} \\ &= 3 \times 144\text{ in}^2 \\ &= 432\text{ in}^2 \end{aligned}

The two fractions together are \displaystyle \left(\frac{12\text{ in}}{1\text{ ft}}\right)^2 = \frac{144\text{ in}^2}{1\text{ ft}^2}, which still equals 1, so a single multiplication by 144 does the same job. The unit to be canceled goes in the denominator, so going from square inches to square feet the fraction is turned over and the amount is divided by 144.

This gives a sense check too: a smaller unit fits more times into the same area, so changing to a smaller unit makes the number larger, and changing to a larger unit makes it smaller.

Converting the lengths first

When the lengths are known, there is a second route: change each length to the new unit, then find the area or volume from the new lengths. A rectangle 2 ft by 3 ft is 24 in by 36 in, so its area is 24 \times 36 = 864\text{ in}^2, which agrees with 6 \times 144 = 864. The two routes always agree, because each length carries one factor of 12 into the product, so one route can check the other.

The facts used most

Length	Area	Volume
1\text{ cm} = 10\text{ mm}	1\text{ cm}^2 = 100\text{ mm}^2	1\text{ cm}^3 = 1{,}000\text{ mm}^3
1\text{ m} = 100\text{ cm}	1\text{ m}^2 = 10{,}000\text{ cm}^2	1\text{ m}^3 = 1{,}000{,}000\text{ cm}^3
1\text{ km} = 1{,}000\text{ m}	1\text{ km}^2 = 1{,}000{,}000\text{ m}^2
1\text{ ft} = 12\text{ in}	1\text{ ft}^2 = 144\text{ in}^2	1\text{ ft}^3 = 1{,}728\text{ in}^3
1\text{ yd} = 3\text{ ft}	1\text{ yd}^2 = 9\text{ ft}^2	1\text{ yd}^3 = 27\text{ ft}^3

Two units have names of their own. A hectare, written \text{ha}, is the area of a square 100 m on each side, so 1\text{ ha} = 100 \times 100 = 10{,}000\text{ m}^2, and since a square kilometer is 10 \times 10 such squares, 1\text{ km}^2 = 100\text{ ha}. A liter, written \text{L}, is the volume of a cube 10 cm on each edge, so 1\text{ L} = 10 \times 10 \times 10 = 1{,}000\text{ cm}^3. A milliliter is a thousandth of a liter, which makes it exactly one cubic centimeter (see liters and milliliters). Flow rate builds the cubic meter out of these liter cubes.

Worked examples

Example 1

Carpet by the square yard

Imelda’s sitting room is a rectangle 14 ft by 13.5 ft. Carpet is sold by the square yard at 38 dollars a square yard. How many square yards cover the floor, and what do they cost?

Solution

The area in square feet is length times width:

14 \times 13.5 = 189\text{ ft}^2

A square yard is a square 3 ft on each side, so it holds 3 \times 3 = 9 square feet. Going to the larger unit, the number gets smaller, so divide:

189 \div 9 = 21\text{ yd}^2

The carpet costs 21 \times 38 = 798 dollars.

Check. Convert the lengths first. Each length in feet is divided by 3 once: \displaystyle 14\text{ ft} = \frac{14}{3}\text{ yd} and 13.5\text{ ft} = 4.5\text{ yd}, and \displaystyle \frac{14}{3} \times 4.5 = \frac{63}{3} = 21. Dividing 189 by 3 only once would have changed one side of the room and left the other in feet.

Example 2

Metric areas in both directions

Convert (a) 3.6\text{ m}^2 to square centimeters, (b) 4{,}500\text{ cm}^2 to square meters, (c) 2.5\text{ cm}^2 to square millimeters.

Solution

(a) A square meter is 100 cm by 100 cm, so it holds 100^2 = 10{,}000 square centimeters. The centimeter is the smaller unit, so the number grows:

3.6 \times 10{,}000 = 36{,}000\text{ cm}^2

(b) Now the change is to the larger unit, so divide by the same 10{,}000:

4{,}500 \div 10{,}000 = 0.45\text{ m}^2

Dividing by 10{,}000 moves the decimal point four places, two for each of the two lengths.

2.5 \times 100 = 250\text{ mm}^2

Example 3

A field in hectares

Horatio’s field is a rectangle 320 m by 185 m. Find its area in square meters, in hectares and in square kilometers.

Solution

The area is

320 \times 185 = 59{,}200\text{ m}^2

A hectare is a square 100 m on each side, which is 10{,}000\text{ m}^2, so

59{,}200 \div 10{,}000 = 5.92\text{ ha}

A square kilometer is 1{,}000 m by 1{,}000 m, which is 1{,}000{,}000\text{ m}^2, so

59{,}200 \div 1{,}000{,}000 = 0.0592\text{ km}^2

Check. A square kilometer is 100 hectares, and 5.92 \div 100 = 0.0592. The hectare gives a size you can picture: a little under six squares each 100 m on a side.

Example 4

Paint for a wall measured in centimeters

Gwendolyn measures a wall at 480 cm by 265 cm. She will give it two coats of paint, and one liter of the paint covers 12\text{ m}^2. The paint is sold in 1-liter tins. How much paint does she need, and how many tins?

Solution

The wall’s area in square centimeters is 480 \times 265 = 127{,}200\text{ cm}^2. The coverage is in square meters, so the area must be in square meters too. Each square meter is 10{,}000 square centimeters:

127{,}200 \div 10{,}000 = 12.72\text{ m}^2

Two coats cover the wall twice, so the paint must cover 2 \times 12.72 = 25.44\text{ m}^2. Each liter covers 12\text{ m}^2:

25.44 \div 12 = 2.12\text{ L}

Two tins hold only 2 liters, so she needs 3 tins.

Check. Convert the lengths first: 480\text{ cm} = 4.8\text{ m} and 265\text{ cm} = 2.65\text{ m}, and 4.8 \times 2.65 = 12.72\text{ m}^2, the same area.

Example 5

Concrete by the cubic yard

Jeremiah is pouring a patio slab 18 ft long, 12 ft wide and 4 in thick. Concrete is ordered in whole cubic yards. Find the volume in cubic feet and in cubic yards, and how much to order.

Solution

The three lengths must share a unit before they are multiplied. Since 12\text{ in} = 1\text{ ft}, the thickness is \displaystyle 4\text{ in} = \frac{4}{12}\text{ ft} = \frac{1}{3}\text{ ft}. The slab is a box, so

18 \times 12 \times \frac{1}{3} = 72\text{ ft}^3

A cubic yard is 3 \times 3 \times 3 = 27 cubic feet, and the yard is the larger unit, so divide:

72 \div 27 = \frac{8}{3} = 2\tfrac{2}{3}\text{ yd}^3 \approx 2.67\text{ yd}^3

Two cubic yards would leave the slab short, so Jeremiah orders 3 cubic yards.

Check. Convert the lengths to yards first. 18\text{ ft} = 6\text{ yd}, 12\text{ ft} = 4\text{ yd}, and since 1\text{ yd} = 36\text{ in}, \displaystyle 4\text{ in} = \frac{4}{36} = \frac{1}{9}\text{ yd}. Then \displaystyle 6 \times 4 \times \frac{1}{9} = \frac{24}{9} = \frac{8}{3}, as before.

Example 6

A fish tank in liters

Katarzyna’s fish tank is a box measured inside as 0.8 m long, 35 cm wide and 400 mm high. How many liters does it hold when full? She fills it to a depth of 36 cm. How many liters of water is that?

Solution

The three lengths are in three different units, so first put them in one. Centimeters suit the liter, since 1\text{ L} = 1{,}000\text{ cm}^3. With 1\text{ m} = 100\text{ cm} and 1\text{ cm} = 10\text{ mm}, the tank is 80 cm long, 35 cm wide and 40 cm high. Full, it holds

80 \times 35 \times 40 = 112{,}000\text{ cm}^3

and each liter is 1{,}000\text{ cm}^3, so that is 112{,}000 \div 1{,}000 = 112 liters.

The water is a box with the same base and a height of 36 cm:

80 \times 35 \times 36 = 100{,}800\text{ cm}^3

which is 100{,}800 \div 1{,}000 = 100.8 liters.

Check. A liter is a cube 10 cm on each edge, which is a cubic decimeter. In decimeters the tank is 8 by 3.5 by 4, and 8 \times 3.5 \times 4 = 112 cubic decimeters, so 112 liters.

Boundaries

Square the factor only for an area, cube it only for a volume. The power matches the number of lengths multiplied together in the unit. Cloth sold by the meter off a roll of fixed width is a length, so it converts with the length factor once, even though it covers a surface.

Areas convert to areas and volumes to volumes, never across. Square meters and cubic meters measure different things, so no factor turns one into the other. Going from an area to a volume needs a third length, such as the slab’s thickness above.

A rounded length factor gets worse when it is squared or cubed. Since 1\text{ in} = 2.54\text{ cm} exactly, 1\text{ in}^2 = 2.54^2 = 6.4516\text{ cm}^2 exactly. A rounded factor such as 1\text{ m} \approx 3.28\text{ ft} is off by about 0.03\% as a length, but cubed it is off by about three times as much, so keep the exact factor until the end.

Some units are not a length squared or cubed. The liter (1{,}000\text{ cm}^3) and the hectare (10{,}000\text{ m}^2) still fit the metric family, but an acre is 43{,}560\text{ ft}^2 and a United States gallon is 231\text{ in}^3. Those are facts to look up, not a factor to square or cube.

Where this goes next

Rates with an area or a volume in them change units the same way, one factor for each length. Modeling with solids turns a density in grams per cubic centimeter into kilograms per cubic meter by cubing 100 for the length and dividing by 1{,}000 for the mass. Changing the units of a rate treats such changes in general, and multi-step problems with unit conversions put a conversion in the middle of a longer calculation. Surface area and the volume formulas for solids produce answers in square and cubic units that often need changing before they are used.

Very large and very small conversions, such as 1\text{ km}^2 = 10^{10}\text{ cm}^2, are written most easily in scientific notation, where squaring and cubing a power of ten just doubles or triples its exponent.

Exercises

Metric areas

Convert 7\text{ m}^2 to square centimeters.
Convert 2{,}350\text{ cm}^2 to square meters.
Convert 6.4\text{ cm}^2 to square millimeters.
Convert 3\text{ km}^2 to square meters and to hectares.

Customary areas

Convert 5\text{ ft}^2 to square inches.
Convert 216\text{ ft}^2 to square yards.
Convert 2.5\text{ yd}^2 to square feet.
How many square inches are in one square yard?

Volume and capacity

Convert 4\text{ m}^3 to cubic centimeters.
Convert 54\text{ ft}^3 to cubic yards.
Convert 2\text{ ft}^3 to cubic inches.
Convert 3{,}750\text{ cm}^3 to liters.
Convert 0.6 L to cubic centimeters.
How many cubic millimeters are in one cubic meter?

Problems

Lucius’s bedroom floor is a rectangle 16.5 ft by 12 ft. Carpet costs 30 dollars a square yard. How many square yards does the floor need, and what does the carpet cost?
Gwendolyn’s field is a rectangle 250 m by 140 m. Find its area in hectares. Grass seed is sown at 40 kg per hectare. How much seed does the field need?
A mural is to be painted on a wall section 280 cm by 175 cm, with two coats. One liter of paint covers 7\text{ m}^2. How many liters are needed?
A garden path is 36 ft long and 4.5 ft wide, and its concrete is 4 in thick. How many cubic yards of concrete does it take?
Horatio’s fish tank has a rectangular base 75 cm by 30 cm. How deep is the water when the tank holds 54 liters?
One shop sells a carpet at 36 dollars a square yard and another sells the same carpet at 4.25 dollars a square foot. Which is cheaper?
How many square tiles 6 in on each side have a total area of 15\text{ ft}^2?
One inch is exactly 2.54 cm. Find the exact number of square centimeters in a square inch and of cubic centimeters in a cubic inch.

Reasoning

Use a picture to explain why 1\text{ m}^2 is 10{,}000\text{ cm}^2 and not 100\text{ cm}^2.
To convert 5\text{ yd}^3 to cubic feet, Jeremiah multiplies by 3, then by 3, then by 3. Katarzyna multiplies by 27 once. Explain why each of them is allowed to do this, and why they must get the same answer.
Imelda’s plot is “200 meters square” and Lucius’s is “200 square meters”. Explain the difference, and how many times larger the first plot is.

Answers

70{,}000\text{ cm}^2. A square meter is 100^2 = 10{,}000\text{ cm}^2, and 7 \times 10{,}000 = 70{,}000.
0.235\text{ m}^2. Going to the larger unit, divide: 2{,}350 \div 10{,}000 = 0.235.
640\text{ mm}^2. A square centimeter is 10^2 = 100\text{ mm}^2, and 6.4 \times 100 = 640.
3{,}000{,}000\text{ m}^2, which is 300 ha. A square kilometer is 1{,}000^2 = 1{,}000{,}000\text{ m}^2, and 3{,}000{,}000 \div 10{,}000 = 300.
720\text{ in}^2. A square foot is 12^2 = 144\text{ in}^2, and 5 \times 144 = 720.
24\text{ yd}^2. A square yard is 3^2 = 9\text{ ft}^2, and 216 \div 9 = 24.
22.5\text{ ft}^2. Going to the smaller unit, multiply: 2.5 \times 9 = 22.5.
1{,}296\text{ in}^2. A yard is 36 in, so a square yard is 36^2 = 1{,}296\text{ in}^2; or 9 \times 144 = 1{,}296.
4{,}000{,}000\text{ cm}^3. A cubic meter is 100^3 = 1{,}000{,}000\text{ cm}^3.
2\text{ yd}^3. A cubic yard is 3^3 = 27\text{ ft}^3, and 54 \div 27 = 2.
3{,}456\text{ in}^3. A cubic foot is 12^3 = 1{,}728\text{ in}^3, and 2 \times 1{,}728 = 3{,}456.
3.75 L. A liter is 1{,}000\text{ cm}^3, and 3{,}750 \div 1{,}000 = 3.75.
600\text{ cm}^3. Multiply by 1{,}000: 0.6 \times 1{,}000 = 600, which is also 600 mL.
1{,}000{,}000{,}000\text{ mm}^3. A meter is 1{,}000 mm, so a cubic meter is 1{,}000^3 = 10^9 cubic millimeters.
22\text{ yd}^2, costing 660 dollars. The area is 16.5 \times 12 = 198\text{ ft}^2, and 198 \div 9 = 22; then 22 \times 30 = 660.
3.5 ha, needing 140 kg of seed. The area is 250 \times 140 = 35{,}000\text{ m}^2, and 35{,}000 \div 10{,}000 = 3.5; then 3.5 \times 40 = 140.
1.4 L. The area is 280 \times 175 = 49{,}000\text{ cm}^2 = 4.9\text{ m}^2, two coats make 9.8\text{ m}^2, and 9.8 \div 7 = 1.4.
2\text{ yd}^3. The thickness is \displaystyle 4\text{ in} = \frac{1}{3}\text{ ft}, so the volume is \displaystyle 36 \times 4.5 \times \frac{1}{3} = 54\text{ ft}^3, and 54 \div 27 = 2.
24 cm. The water is 54 \times 1{,}000 = 54{,}000\text{ cm}^3, the base is 75 \times 30 = 2{,}250\text{ cm}^2, and 54{,}000 \div 2{,}250 = 24.
The first shop. A square yard is 9 square feet, so 36 dollars a square yard is 36 \div 9 = 4 dollars a square foot, less than 4.25 dollars. (Or: 4.25 dollars a square foot is 9 \times 4.25 = 38.25 dollars a square yard.)
60 tiles. One tile is 6 \times 6 = 36\text{ in}^2, and 15\text{ ft}^2 = 15 \times 144 = 2{,}160\text{ in}^2, so 2{,}160 \div 36 = 60. Or: four tiles make a square foot, and 4 \times 15 = 60.
6.4516\text{ cm}^2 and 16.387064\text{ cm}^3. Square and cube the length factor: 2.54^2 = 6.4516 and 2.54^3 = 16.387064.
Draw a square 1 m on each side and mark each side in 100 centimeters. Lines through the marks cut the square into 100 rows, each of 100 squares 1 cm on a side, so there are 100 \times 100 = 10{,}000 of them. A row of 100 square centimeters is only a strip 1 cm wide along one side.
Each \displaystyle \frac{3\text{ ft}}{1\text{ yd}} equals 1, so multiplying by it does not change the volume; a cubic yard is a yard three times over, so it takes three of them to cancel every yard. Multiplying by 3 three times is multiplying by 3 \times 3 \times 3 = 27, by the associative property, so both give 135\text{ ft}^3.
“200 meters square” is a square 200 m on each side, with area 200 \times 200 = 40{,}000\text{ m}^2, or 4 ha. “200 square meters” is an area of 200\text{ m}^2, such as a square about 14 m on a side. The first is 40{,}000 \div 200 = 200 times larger.